Differentiable vector-valued functions from Euclidean space: Difference between revisions

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as <math>i,</math> <math>\ell,</math> and <math>\alpha \in A</math> vary in all possible ways.{{sfn|Trèves|2006|pp=412–419}}

If <math>\Omega</math> is a compact space and <math>Y</math> is a Banach space, then <math>C^0(\Omega;Y)</math> becomes a Banach space normed by <math>\| f \| := \sup_{\omega \in \Omega} \| f(\omega) \|.</math>{{sfn|Trèves|2006|pp=446–451}}

=== Space of ''C''^''k'' functions with support in a compact subset ===

The definition of the topology of the [[space of test functions]] is now duplicated and generalized.

For any compact subset <math>K \subseteq \Omega,</math> let \right) denote the set of all <math>f</math> in <math>C^k(\Omega;Y)</math> whose support lies in <math>K</math> (in particular, if <math>f \in C^k(K;Y)</math> then the ___domain of <math>f</math> is <math>\Omega</math> rather than <math>K</math>) and give \right)it the subspace topology induced by <math>C^k(\Omega;Y).</math>{{sfn|Trèves|2006|pp=412–419}}

If <math>\OmegaK</math> is a compact space and <math>Y</math> is a Banach space, then <math>C^0(\OmegaK;Y)</math> becomes a Banach space normed by <math>\| f \| := \sup_{\omega \in \Omega} \| f(\omega) \|.</math>{{sfn|Trèves|2006|pp=446–451}}

Let <math>C^k(K)</math> denote <math>C^k\left(K; \mathbb{F}\right).</math>

Note that forFor any two compact subsets <math>K \subseteq L \subseteq \Omega,</math> the inclusion
:<math>\operatorname{In}_{K}^{L} : C^k(K;Y) \to C^k(L;Y)</math>
is an embedding of TVSs and that the union of all <math>C^k(K;Y),</math> as <math>K</math> varies over the compact subsets of <math>\Omega,</math> is <math>C_c^k(\Omega;Y).</math>

=== Space of compactly support ''C''^''k'' functions ===

For any compact subset <math>K \subseteq \Omega,</math> let
:<math>\operatorname{In}_K : C^k(K;Y) \to C_c^k(\Omega;Y)</math> be
denote the inclusion map and giveendow <math>C_c^k(\Omega;Y)</math> with the strongest topology making all <math>\operatorname{In}_K</math> continuous, which is known as the [[final topology]] induced by these map.

The spaces <math>C^k(K;Y)</math> and maps <math>\operatorname{In}_{K_1}^{K_2}</math> form a [[direct limit|direct system]] (directed by the compact subsets of <math>\Omega</math>) whose limit in the category of TVSs is <math>C_c^k(\Omega;Y)</math> together with the injections <math>\operatorname{In}_{K}.</math>{{sfn|Trèves|2006|pp=412–419}}

The spaces <math>C^k\left(\overline{\Omega_i}; Y\right)</math> and maps <math>\operatorname{In}_{\overline{\Omega_i}}^{\overline{\Omega_j}}</math> also form a [[direct limit|direct system]] (directed by the total order <math>\mathbb{N}</math>) whose limit in the category of TVSs is <math>C_c^k(\Omega;Y)</math> together with the injections <math>\operatorname{In}_{\overline{\Omega_i}}.</math>{{sfn|Trèves|2006|pp=412–419}}

Each embedding <math>\operatorname{In}_K</math> is an embedding of TVSs.

A subset <math>S</math> of <math>C_c^k(\Omega;Y)</math> is a neighborhood of the origin in <math>C_c^k(\Omega;Y)</math> if and only if <math>S \cap C^k(K;Y)</math> is a neighborhood of the origin in <math>C^k(K;Y)</math> for every compact <math>K \subseteq \Omega.</math>

This direct limit topology (i.e. the final topology) on <math>C_c^\infty(\Omega)</math> is known as the '''{{em|canonical LF topology}}'''.

If <math>Y</math> is a Hausdorff locally convex space, <math>T</math> is a TVS, and <math>u : C_c^k(\Omega;Y) \to T</math> is a linear map, then <math>u</math> is continuous if and only if for all compact <math>K \subseteq \Omega,</math> the restriction of <math>u</math> to <math>C^k(K;Y)</math> is continuous.{{sfn|Trèves|2006|pp=412–419}} OneThe replacestatement remains true if "all compact <math>K \subseteq \Omega</math>" is replaced with "all <math>K := \overline{\Omega}_i</math>".

=== Properties ===

{{mathMath theorem|name=Theorem{{sfn|Trèves|2006|pp=412–419}}|note=|style=|math_statement=

Let <math>m</math> be a positive integer and let <math>\Delta</math> be an open subset of <math>\R^m.</math>

Given <math>\phi \in C^k(\Omega \times \Delta),</math> for any <math>y \in \Delta</math> let <math>\phi_y : \Omega \to \mathbb{F}</math> be defined by <math>\phi_y(x) = \phi(x, y);</math> and let <math>I_k(\phi) : \Delta \to C^k(\Omega)</math> be defined by <math>I_k(\phi)(y) := \phi_y.</math>

Then

and let <math>I_k(\phi) : \Delta \to C^k(\Omega)</math> be defined by <math>I_k(\phi)(y) := \phi_y.</math>

Then :<math>I_\infty : C^\infty(\Omega \times \Delta) \to C^\infty(\Delta; C^\infty(\Omega))</math> is a (surjective) isomorphism of TVSs.

Furthermore, its restriction

Furthermore, the restriction :<math>I_{\infty}\big\vert_{C_c^{\infty}\left(\Omega \times \Delta\right)} : C_c^\infty(\Omega \times \Delta) \to C_c^\infty\left(\Delta; C_c^\infty(\Omega)\right)</math> is an isomorphism of TVSs when <math>C_c^\infty\left(\Omega \times \Delta\right)</math> has its canonical LF topology.

is an isomorphism of TVSs when <math>C_c^\infty\left(\Omega \times \Delta\right)</math> has its canonical LF topology.

}}

{{mathMath theorem|name=Theorem{{sfn|Trèves|2006|pp=412-419}}|note=|style=|math_statement=

ForLet <math>Y</math> be a Hausdorff [[Locally convex topological vector space|locally convex]] [[topological vector space]] and for every continuous linear form <math>y^{\prime} \in Y</math> and every <math>f \in C^\infty(\Omega;Y),</math> let <math>J_{y^{\prime}}(f) : \Omega \to \mathbb{F}</math> be defined by <math>J_{y^{\prime}}(f)(p) = y^{\prime}(f(p)).</math>

Let <math>Y</math> be a Hausdorff locally convex space.

Then

For every continuous linear form <math>y^{\prime} \in Y</math> and every <math>f \in C^\infty(\Omega;Y),</math> let <math>J_{y^{\prime}}(f) : \Omega \to \mathbb{F}</math> be defined by <math>J_{y^{\prime}}(f)(p) = y^{\prime}(f(p)).</math>

Then :<math>J_{y^{\prime}} : C^\infty(\Omega;Y) \to C^\infty(\Omega)</math>
is a continuous linear map;

and furthermore, theits restriction
:<math>J_{y^{\prime}}\big\vert_{C_c^\infty(\Omega;Y)} : C_c^\infty(\Omega;Y) \to C^\infty(\Omega)</math>
is also continuous (where <math>C_c^\infty(\Omega;Y)</math> has the canonical LF topology).

}}